Rings, Commutative Rings, Idempotents, Subrings and Isomorphisms

(1) Given a ring R, an element e is called an idempotent if e^2 = e.
(i) Let R1 and R2 be two commutative rings with unity. Consider R = R1 x R2. Find two non-zero idempotents e1,e2 E R such that 1= e1 + e2 and e1e2 = 0. (Be careful: what is 1 in this R? What is 0?)
(ii) On the other hand, suppose R is any commutative ring with unity and e1 E R is an idempotent. Show that ... is also an idempotent with....
(iii) Given R a commutative ring with unity and e1 e R a nonzero idempotent. Show that R1 =... is a subring of R with unity. Warning: the unity of R1 may not be the unity of I?. In fact, guess what the unity of R1 is, then prove your guess.

(2) Let : R1 ?4 R be a ring isomorphism. Define ....
(ii) Show that ... is irreducible if and only if ... is irreducible.

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Rings, Commutative Rings, Idempotents, Subrings and Isomorphisms are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.